## Water and Wheelchairs: The 2006 Mathematical Contest in Modeling

**September 24, 2006**

Ably representing the team from the University of Colorado, Boulder, Bradley Klingenberg single-handedly presented the team’s SIAM award-winning solution to the continuous (A) problem in MCM 06 and accepted the prize on behalf of his team at the awards lunch. Not pictured are team members Brian Camley and Pascal Getreuer.

James Case

Seven hundred forty-seven student teams---a new record---participated in the 2006 Mathematical Contest in Modeling. It was the twenty-second year the contest had been held, and the second in which a majority of the teams were from China. The problems were posted on the Web on Thursday, February 2, at 8 PM EST; solutions were to be submitted by the evening of Monday, February 6, at the same time.

Five hundred sixteen of the three-member teams elected to work on problem A, the continuous problem. Six of their solutions were judged to be "outstanding," 85 were deemed "meritorious," and 132 received "honorable mention." The other two hundred thirty-one teams chose to work on problem B, the discrete alternative. Five of the solutions from this group were deemed outstanding, 37 were designated meritorious, and 56 received honorable mention.

The students who chose the A problem were asked to design an irrigation system for a relatively small (family) farm or ranch. As the students were told, modern irrigation systems are constructed from lightweight aluminum pipes and sprinkler heads. Self-propelled systems are expensive to purchase but cheap to operate, while systems that must be manually repositioned from time to time--so that each part of the field receives an appropriate quantity of water--are cheap to purchase and expensive to operate. The students were asked to configure a system and a repositioning schedule capable of delivering at minimum cost at least 2 centimeters of water every four days to every part of a field 30 meters wide and 80 meters long, without exceeding 0.75 cm per hour of water at any location. Available for the purpose was a single water source with a flow rate of 150 liters per minute at the wellhead, under 420 kilopascals of pressure. The inner diameter of each pipe was to be 10 cm, and that of each rotating spray nozzle 0.6 cm.

Problem B solvers were to advise the fictitious Epsilon Airlines on the operation of wheelchair fleets at airports. An act of the U.S. Congress requires that every commercial airline provide free wheelchair service to disabled passengers, and all airlines wish to minimize the cost of compliance. Each must therefore decide how many wheelchairs to purchase, and how many of them to staff, at every hour of every day, at every airport served. Along with the purchase price of wheelchairs and the wages of escorts and dispatchers, the students were asked to consider a wide variety of operating costs--including maintenance, storage, and liability costs, and the costs of delaying flights until wheelchair-dependent passengers can be brought onboard--in advising Epsilon Airlines on efficient management of the fleets it must operate at each and every airport it serves.

For the B problem---which this reporter helped to judge---the stronger teams recognized from the outset that the heart of Epsilon's problem lay in the design of the algorithm needed to dispatch wheelchairs and "escorts" to appropriate gates, incorporating a reasonable estimate of the effect of underperformance on market share. A number of the other teams got caught up in such ancillary issues as the make, model, and maintenance schedule of the wheelchairs to be provided, or of the cell phones with which each escort would naturally be equipped. The wheelchair dispatcher's problem differs from that confronting, say, a taxicab company in several respects: The ETAs of most (though by no means all) wheelchair-dependent passengers are known well in advance, and the impact of late arrival on market share is significantly greater, in that people tend to hold grudges against servers they believe to have "mistreated grandma." These and other features of the problem, as one team paused to observe, complicate the task of "making Epsilon greater than Delta."

The outstanding papers on the A problem came from the University of Colorado at Boulder, Zhejiang University of Technology, Shanghai Jiao Tong University, Duke University, the University of California at Davis, and Carroll College in Helena, Montana; the MAA and SIAM awards went to team 155 from Boulder, and the INFORMS award to team 783 from Duke. For the B problem, the papers judged outstanding were from Harvard, MIT, Rice, Harvard again, and Rensselaer Polytechnic Institute; the MAA award went to team 868, from Rice, the INFORMS award to team 609 from MIT, and the SIAM award to team 806 from the math department at Harvard.

The Ben Fusaro Award---named for the contest founder---recognizes the paper on each problem that best meets the following criteria:

- The paper presents high-quality application of the complete modeling process, as represented by the 100-point scale developed for the final round of judging.
- The team has demonstrated noteworthy originality and creativity in their modeling effort to solve the problem as given.
- The paper is well written, in a clear expository style, and is a pleasure to read.

The Fusaro Awards went to team 600, from Shanghai Jiao Tong University, for problem A, and to team 887, from Maggie Walker Governor's School in Richmond, Virginia, for problem B.

*SIAM president Martin Golubitsky congratulated Harvard University team members Benjamin Conlee, Neal Gupta, and Christopher Yetter, who received the SIAM Award in the MCM for their solution to problem B, the discrete problem.*

Both SIAM winners presented their solutions at a special session of the 2006 SIAM Annual Meeting in Boston. All three members of the Harvard team participated in that presentation, while the Colorado team was ably represented by Bradley Klingenberg alone. The former concentrated on assessing the impact of poor wheelchair performance on Epsilon Airlines' market share; the latter appeared to address every aspect of problem A with equal thoroughness. All in attendance---including past, present, and aspiring MCM team coaches, and at least one member of an opposing team---were impressed by the talent on display.

*James Case writes from Baltimore, Maryland.*